3.2826 \(\int \frac{1}{\sqrt{2-x} \sqrt{1+x} \sqrt{3+x}} \, dx\)

Optimal. Leaf size=24 \[ \sqrt{2} F\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right )|-\frac{3}{2}\right ) \]

[Out]

Sqrt[2]*EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], -3/2]

_______________________________________________________________________________________

Rubi [A]  time = 0.0407108, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \sqrt{2} F\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right )|-\frac{3}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 - x]*Sqrt[1 + x]*Sqrt[3 + x]),x]

[Out]

Sqrt[2]*EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], -3/2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.98597, size = 24, normalized size = 1. \[ \sqrt{2} F\left (\operatorname{asin}{\left (\frac{\sqrt{3} \sqrt{x + 1}}{3} \right )}\middle | - \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2-x)**(1/2)/(1+x)**(1/2)/(3+x)**(1/2),x)

[Out]

sqrt(2)*elliptic_f(asin(sqrt(3)*sqrt(x + 1)/3), -3/2)

_______________________________________________________________________________________

Mathematica [B]  time = 0.112191, size = 67, normalized size = 2.79 \[ -\frac{2 (x+3) \sqrt{1-\frac{5}{x+3}} \sqrt{1-\frac{2}{x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{5}}{\sqrt{x+3}}\right )|\frac{2}{5}\right )}{\sqrt{-5 (x+3)^2+35 (x+3)-50}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[2 - x]*Sqrt[1 + x]*Sqrt[3 + x]),x]

[Out]

(-2*(3 + x)*Sqrt[1 - 5/(3 + x)]*Sqrt[1 - 2/(3 + x)]*EllipticF[ArcSin[Sqrt[5]/Sqr
t[3 + x]], 2/5])/Sqrt[-50 + 35*(3 + x) - 5*(3 + x)^2]

_______________________________________________________________________________________

Maple [B]  time = 0.099, size = 44, normalized size = 1.8 \[ -{\frac{\sqrt{3}}{3+3\,x}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2-2\,x}},{\frac{i}{3}}\sqrt{3}\sqrt{2} \right ) \sqrt{-2-2\,x}\sqrt{2+2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2-x)^(1/2)/(1+x)^(1/2)/(3+x)^(1/2),x)

[Out]

-1/3*EllipticF(1/2*(-2-2*x)^(1/2),1/3*I*3^(1/2)*2^(1/2))*3^(1/2)*(-2-2*x)^(1/2)*
(2+2*x)^(1/2)/(1+x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 3} \sqrt{x + 1} \sqrt{-x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x + 3} \sqrt{x + 1} \sqrt{-x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x + 2} \sqrt{x + 1} \sqrt{x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2-x)**(1/2)/(1+x)**(1/2)/(3+x)**(1/2),x)

[Out]

Integral(1/(sqrt(-x + 2)*sqrt(x + 1)*sqrt(x + 3)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 3} \sqrt{x + 1} \sqrt{-x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)), x)